Local Poincar\'e Algebra from Quantum Chaos

TL;DR

This paper reveals a universal emergence of a local Poincaré algebra near horizons in quantum gravity, linking quantum chaos, modular theory, and thermodynamics, with implications for holography and black hole physics.

Contribution

It demonstrates that a two-dimensional Poincaré algebra arises in quantum systems with modular subalgebras, establishing a connection between quantum chaos, modular flow, and horizon symmetries.

Findings

Modular future/past subalgebras imply a second law of modular thermodynamics.

Quantum K-systems are always maximally chaotic.

Exponential decay of modular correlators is proven in these systems.

Abstract

The local two-dimensional Poincar\'e algebra near the horizon of an eternal AdS black hole, or in proximity to any bifurcate Killing horizon, is generated by the Killing flow and outward null translations on the horizon. In holography, this local Poincar\'e algebra is reflected as a pair of unitary flows in the boundary Hilbert space whose generators under modular flow grow and decay exponentially with a maximal Lyapunov exponent. This is a universal feature of many geometric vacua of quantum gravity. To explain this universality, we show that a two-dimensional Poincar\'e algebra emerges in any quantum system that has von Neumann subalgebras associated with half-infinite modular time intervals (modular future and past subalgebras) in a limit analogous to the near-horizon limit. In ergodic theory, quantum dynamical systems with future or past algebras are called quantum K-systems. The surprising statement is that modular K-systems are always maximally chaotic. Interacting quantum systems in the thermodynamic limit and large $N$ theories above the Hawking-Page phase transition are examples of physical theories with future/past subalgebras. We prove that the existence of (modular) future/past von Neumann subalgebras also implies a second law of (modular) thermodynamics and the exponential decay of (modular) correlators. We generalize our results from the modular flow to any dynamical flow with a positive generator and interpret the positivity condition as quantum detailed balance.